Given a topological space $X$, ${\rm aut}(X)$ denotes the monoid of the homotopy self equivalences of $X$, that are maps $f: X\rightarrow X$ which admits a homotopy inverse. ${\rm aut}_1(X)$ denotes the path component of the identity map. The homotopical niloptency of ${\rm aut}_1(X)$ as H-space, denoted here ${\rm Hnil} ({\rm aut}_1(X))$ is then the least integer $n$ such that $(n+1)$-th commutator is nullhomotopic.

**Question**: When $G$ is a connected topological group, it is true that $${\rm Nil}(G)\leq {\rm Hnil} ({\rm aut}_1(G)) ?$$